# Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs

arXiv: https://arxiv.org/abs/2208.01565 · HTML: https://mlsanity.com/paper/2208.01565/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 73 references that are in the map.
- First posted on arXiv 2 Aug 2022.

## Builds on (most cited first, 30 of 37)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Why ReLU networks yield high-confidence predictions far away from the training data and how to mitigate the problem](https://mlsanity.com/paper/1812.05720/) (2018)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks](https://mlsanity.com/paper/2002.10118/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning](https://mlsanity.com/paper/1902.03932/) (2019)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning](https://mlsanity.com/paper/2104.04975/) (2021)
- [Noisy Natural Gradient as Variational Inference](https://mlsanity.com/paper/1712.02390/) (2017)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)

## Built on by (most cited first, 25 of 25)

- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'](https://mlsanity.com/paper/2505.04950/) (2025)
- [Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators](https://mlsanity.com/paper/2508.00643/) (2025)
- [Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation](https://mlsanity.com/paper/2402.15656/) (2024)
- [Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equations](https://mlsanity.com/paper/2404.15618/) (2024)
- [Locally Adaptive Conformal Inference for Operator Models](https://mlsanity.com/paper/2507.20975/) (2025)
- [Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogates](https://mlsanity.com/paper/2603.11052/) (2026)
- [ON-Traffic: An Operator Learning Framework for Online Traffic Flow Estimation and Uncertainty Quantification from Lagrangian Sensors](https://mlsanity.com/paper/2503.14053/) (2025)
- [Neural Measures for learning distributions of Random PDEs](https://mlsanity.com/paper/2507.01687/) (2025)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026)
- [Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning](https://mlsanity.com/paper/2606.17513/) (2026)
- [Neural Operator Processes for Probabilistic Operator Learning under Partial Observations](https://mlsanity.com/paper/2606.22946/) (2026)
- [GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models](https://mlsanity.com/paper/2608.21652/) (2026)
- [Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problems](https://mlsanity.com/paper/2505.21723/) (2025)
- [From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systems](https://mlsanity.com/paper/2506.15906/) (2025)
- [Infinite Neural Operators: Gaussian processes on functions](https://mlsanity.com/paper/2510.16675/) (2025)
- [Generative Neural Operators through Diffusion Last Layer](https://mlsanity.com/paper/2602.04139/) (2026)
- [Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Field](https://mlsanity.com/paper/2605.03247/) (2026)
- [Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization](https://mlsanity.com/paper/2605.15806/) (2026)
- [Conformal Uncertainty Quantification Guarantees for Neural Operators](https://mlsanity.com/paper/2608.28515/) (2026)
- [Threshold-Aware Conformal Routing](https://mlsanity.com/paper/2610.02487/) (2026)

## Abstract (arXiv)

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit uncertainty quantification. This is arguably even more of a problem for this kind of tasks than elsewhere in machine learning, because the dynamical systems typically described by PDEs often exhibit subtle, multiscale structure that makes errors hard to spot by humans. In this work, we first provide a mathematically detailed Bayesian formulation of the ''shallow'' (linear) version of neural operators in the formalism of Gaussian processes. We then extend this analytic treatment to general deep neural operators using approximate methods from Bayesian deep learning. We extend previous results on neural operators by providing them with uncertainty quantification. As a result, our approach is able to identify cases, and provide structured uncertainty estimates, where the neural operator fails to predict well.